Math Problem Statement
Solve the system of equations by converting to a matrix equation and using the inverse of the coefficient matrix:
3x + 5y + 4z = 1
4x - y + 4z = 1
2x + 5y + 3z = 1
Solution
To solve this system of equations using the matrix method and the inverse of the coefficient matrix:
System of equations:
We rewrite it in matrix form: where:
\begin{bmatrix} 3 & 5 & 4 \\ 4 & -1 & 4 \\ 2 & 5 & 3 \end{bmatrix}, \quad \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad \mathbf{b} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}.$$ The solution is obtained by: $$\mathbf{x} = A^{-1} \cdot \mathbf{b}$$ ### Steps: 1. **Find the inverse of $$A$$**. 2. Multiply $$A^{-1}$$ with $$\mathbf{b}$$. Let me compute this.import numpy as np # Define the coefficient matrix A and the right-hand side vector b A = np.array([[3, 5, 4], [4, -1, 4], [2, 5, 3]]) b = np.array([1, 1, 1]) # Calculate the inverse of A A_inv = np.linalg.inv(A) # Solve for x by multiplying A_inv with b x = np.dot(A_inv, b) xarray([-6., -1., 6.])array([-6., -1., 6.])successThe solution to the system of equations is: $$(x, y, z) = (-6, -1, 6)$$ Let me know if you would like a detailed explanation or further questions. Here are 5 related questions to expand your understanding: 1. How can the determinant of the coefficient matrix help identify whether the system has a unique solution? 2. What happens if the coefficient matrix is singular (determinant = 0)? 3. Can this system be solved using other methods, like substitution or elimination? How do they compare to the matrix method? 4. How do you compute the inverse of a matrix manually for small-sized matrices? 5. What are the properties of the solution when the coefficient matrix has dependent rows? **Tip:** The matrix method is efficient for small systems, but for larger systems, numerical approximations (e.g., Gaussian elimination or iterative methods) are often more practical.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Systems of Linear Equations
Matrix Inversion
Formulas
Matrix equation: A · x = b
Solution: x = A⁻¹ · b
Theorems
Properties of Matrix Inverses
Determinants for Matrix Invertibility
Suitable Grade Level
Undergraduate Level
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